Solve the problem.
-A small frictionless cart, attached to the wall by a spring, is pulled $10 \mathrm {~cm}$ back from its rest position and released at time $t = 0$ to roll back and forth for $4 \mathrm { sec }$. Its position at time $t$ is
$\mathrm { s } = 1 - 10 \cos \pi \mathrm { t }$. What is the cart's maximum speed? When is the cart moving that fast? What is the magnitude of of the acceleration then?
A) $10 \pi \approx 31.42 \mathrm {~cm} / \mathrm { sec } ; \mathrm { t } = 0.5 \mathrm { sec } , 1.5 \mathrm { sec } , 2.5 \mathrm { sec } , 3.5 \mathrm { sec }$; acceleration is $0 \mathrm {~cm} / \mathrm { sec } ^ { 2 }$
B) $\pi \approx 3.14 \mathrm {~cm} / \mathrm { sec } ; \mathrm { t } = 0.5 \mathrm { sec } , 1.5 \mathrm { sec } , 2.5 \mathrm { sec } , 3.5 \mathrm { sec }$; acceleration is $0 \mathrm {~cm} / \mathrm { sec } ^ { 2 }$
C) $10 \pi \approx 31.42 \mathrm {~cm} / \mathrm { sec } ; t = 0.5 \mathrm { sec } , 2.5 \mathrm { sec } ;$ acceleration is $1 \mathrm {~cm} / \mathrm { sec } ^ { 2 }$
D) $10 \pi \approx 31.42 \mathrm {~cm} / \mathrm { sec } ; \mathrm { t } = 0 \mathrm { sec } , 1 \mathrm { sec } , 2 \mathrm { sec } , 3 \mathrm { sec }$; acceleration is $0 \mathrm {~cm} / \mathrm { sec } ^ { 2 }$

Question

Quizplus · Accepted Answer

To find the cart's maximum speed, we need to find the velocity function by taking the derivative of the given position function:

$\mathrm{v} = \frac{d\mathrm{s}}{dt} = 10\pi\sin(\pi t)$

The maximum speed occurs when the sine function reaches its maximum or minimum value, which is $\pm1$ respectively. Therefore, the maximum speed is $10\pi \mathrm{~cm/s}$.

The cart is moving that fast at time $t = 0.5$, $1.5$, $2.5$, and $3.5$ seconds. These are the times when the cart passes through the equilibrium point, and its direction changes.

To find the acceleration at the moment of maximum speed, we need to take the derivative of the velocity function:

$\mathrm{a} = \frac{d\mathrm{v}}{dt} = 10\pi^2\cos(\pi t)$

At the moments of maximum speed, the acceleration is zero since the cosine function reaches its maximum or minimum value, which is $\pm1$ respectively, and there is no change in the velocity direction. Therefore, the acceleration is $0 \mathrm{~cm/s^2}$.

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